Econometrics Chengyuan Yin School of Mathematics

Econometrics 21. Two Applications of Maximum Likelihood Estimation and a Two Step Estimation Method

Model for a Binary Dependent Variable Describe a binary outcome. Event occurs or doesn’t (e.g., the democrat wins, the person enters the labor force,… Model the probability of the event Requirements 0 < Probability < 1 P(x) should be monotonic in x – it’s a CDF

Two Standard Models Based on the normal distribution: Prob[y=1|x] = Φ(β’x) = CDF of normal distribution The “probit” model Based on the logistic distribution Prob[y=1|x] = exp(β’x)/[1+ exp(β’x)] The “logit” model Log likelihood P(y|x) = (1-F)(1-y) Fy where F = the cdf Log-L = Σi (1-yi)log(1-Fi) + yilogFi = Σi F[(2yi-1) β’x] since F(-t)=1-F(t) for both.

Coefficients in the Binary Choice Models E[y|x] = 0*(1-Fi) + 1*Fi = P(y=1|x) = F(β’x) The coefficients are not the slopes, as usual in a nonlinear model ∂E[y|x]/∂x= f(β’x) β These will look similar for probit and logit

Application: Female Labor Supply 1975 Survey Data: Mroz (Econometrica) Subsample of the 753 Observations Descriptive Statistics =============================================================================== Variable Mean Std.Dev. Minimum Maximum Cases LFP .600000000 .490880694 .000000000 1.00000000 250 WHRS 799.840000 915.603480 .000000000 4950.00000 250 KL6 .236000000 .511223432 .000000000 3.00000000 250 K618 1.36400000 1.37077353 .000000000 8.00000000 250 WA 42.9200000 8.42648340 30.0000000 60.0000000 250 WE 12.3520000 2.16491186 5.00000000 17.0000000 250 WW 2.27523000 2.59774974 .000000000 14.6310000 250 HHRS 22.3483200 6.00670151 7.68000000 50.1000000 250 HA 45.0240000 8.17132217 30.0000000 60.0000000 250 HE 12.5360000 3.10600920 3.00000000 17.0000000 250 HW 7.49443480 4.63619249 1.08980000 40.5090000 250 FAMINC 23.0625400 12.9239815 3.30500000 91.0440000 250 KIDS .684000000 .465845520 .000000000 1.00000000 250

Marginal Effects

GARCH Models: A Model for Time Series with Latent Heteroscedasticity Bollerslev/Ghysel, 1974

ARCH Model

GARCH Model

Estimated GARCH Model

2 Step Estimation (Murphy-Topel) Setting, fitting a model which contains parameter estimates from another model. Typical application, inserting a prediction from one model into another. A. Procedures: How it's done. B. Asymptotic results: 1. Consistency 2. Getting an appropriate estimator of the asymptotic covariance matrix The Murphy - Topel result Application: Equation 1: Number of children Equation 2: Labor force participation

Setting Two equation model: Procedure: Model for y1 = f(y1 | x1,θ1) Model for y2 = f(y2 | x2, θ2, x1, θ1)) (Note, not ‘simultaneous’ or even ‘recursive.’) Procedure: Estimate θ1 by ML, with covariance matrix (1/n)V1 Estimate θ2 by ML treating θ1 as if it were known. Correct the estimated asymptotic covariance matrix, (1/n)V2 for the estimator of θ2

Murphy and Topel (1984) Results Both MLEs are consistent

M&T Computations

Example Equation 1: Number of Kids - Poisson Regression p(yi1|xi1, β)=exp(-λi)λiyi1/yi1! λi = exp(xi1’β) gi1 = xi1 (yi1 – λi) V1 = [(1/n)Σ(-λi)xi1xi1’]-1

Example - Continued Equation 2: Labor Force Participation – Logit p(yi2|xi2,δ,α,xi1,β)=exp(di2)/[1+exp(di2)]=Pi2 di2 = (2yi2-1)[δ’xi2 + αλi] λi = exp(β’xi1) Let zi2 = (xi2, λi), θ2 = (δ, α) di2 = (2yi2-1)[θ2’zi2] gi2 = (yi2-Pi2)zi2 V2 = [(1/n)Σ{-Pi2(1-Pi2)}zi2zi2’]-1

Murphy and Topel Correction

? Data transformations. Number of kids, scale income variables Create ; Kids = kl6 + k618 ; income = faminc/10000 ; Wifeinc = ww*whrs/1000 $ ? Equation 1, number of kids. Standard Poisson fertility model. ? Fit equation, collect parameters BETA and covariance matrix V1 ? Then compute fitted values and derivatives Namelist ; X1 = one,wa,we,income,wifeinc$ Poisson ; Lhs = kids ; Rhs = X1 $ Matrix ; Beta = b ; V1 = N*VARB $ Create ; Lambda = Exp(X1'Beta); gi1 = Kids - Lambda $ ? Set up logit labor force participation model ? Compute probit model and collect results. Delta=Coefficients on X2 ? Alpha = coefficient on fitted number of kids. Namelist ; X2 = one,wa,we,ha,he,income ; Z2 = X2,Lambda $ Logit ; Lhs = lfp ; Rhs = Z2 $ Calc ; alpha = b(kreg) ; K2 = Col(X2) $ Matrix ; delta=b(1:K2) ; Theta2 = b ; V2 = N*VARB $ ? Poisson derivative of with respect to beta is (kidsi - lambda)´X1 Create ; di = delta'X2 + alpha*Lambda ; pi2= exp(di)/(1+exp(di)) ; gi2 = LFP - Pi2 ? These are the terms that are used to compute R and C. ; ci = gi2*gi2*alpha*lambda ; ri = gi2*gi1$ MATRIX ; C = 1/n*Z2'[ci]X1 ; R = 1/n*Z2'[ri]X1 ; A = C*V1*C' - R*V1*C' - C*V1*R' ; V2S = V2+V2*A*V2 ; V2s = 1/N*V2S $ ? Compute matrix products and report results Matrix ; Stat(Theta2,V2s,Z2)$

Estimated Equation 1: E[Kids] +---------------------------------------------+ | Poisson Regression | | Dependent variable KIDS | | Number of observations 753 | | Log likelihood function -1123.627 | +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| Constant 3.34216852 .24375192 13.711 .0000 WA -.06334700 .00401543 -15.776 .0000 42.5378486 WE -.02572915 .01449538 -1.775 .0759 12.2868526 INCOME .06024922 .02432043 2.477 .0132 2.30805950 WIFEINC -.04922310 .00856067 -5.750 .0000 2.95163126

Two Step Estimator +---------------------------------------------+ | Multinomial Logit Model | | Dependent variable LFP | | Number of observations 753 | | Log likelihood function -351.5765 | | Number of parameters 7 | +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| Characteristics in numerator of Prob[Y = 1] Constant 33.1506089 2.88435238 11.493 .0000 WA -.54875880 .05079250 -10.804 .0000 42.5378486 WE -.02856207 .05754362 -.496 .6196 12.2868526 HA -.01197824 .02528962 -.474 .6358 45.1208499 HE -.02290480 .04210979 -.544 .5865 12.4913679 INCOME .39093149 .09669418 4.043 .0001 2.30805950 LAMBDA -5.63267225 .46165315 -12.201 .0000 1.59096946 +---------+--------------+----------------+--------+---------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Constant 33.1506089 5.41964589 6.117 .0000 WA -.54875880 .07780642 -7.053 .0000 WE -.02856207 .12508144 -.228 .8194 HA -.01197824 .02549883 -.470 .6385 HE -.02290480 .04862978 -.471 .6376 INCOME .39093149 .27444304 1.424 .1543 LAMBDA -5.63267225 1.07381248 -5.245 .0000